Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | C22×2−1+4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Fitting | C22×2−1+4 | C3 |
| socle | C23 | C22×A4 |
| hypercenter | D4 | C22⋊A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1047 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4,3)(6,10)(7,8,15)(11,18,20,17,12,19)(13,14,16);(1,4)(2,3)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18);(5,13,9,15)(6,19,10,20)(7,16,14,8)(11,18,17,12)Polycyclic presentation:
2^2,3,2^5|d^2=h,e^2=h,b^a=bh,d^c=e,e^c=deh,f^c=fg,g^c=fh,e^d=eh36 conjugacy classes — view the full character table.
| maximal subgroup of | #32885⋊C3, #32887⋊C3, #36078⋊C3, … |
| maximal normal subgroup of | #32885⋊C3, #32887⋊C3, #36078⋊C3, … |
| maximal quotient of | #11710590, #11710594, D4×C22⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20121);